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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>As one example, suppose that <span class="process-math">\(N(t)\)</span> is the number of bacteria growing on a plate of nutrients. At the start of the experiment, suppose that there are <span class="process-math">\(1000\)</span> bacteria, so <span class="process-math">\(N(0) = 1000\text{.}\)</span> The rate of change of <span class="process-math">\(N\)</span> will be proportional to <span class="process-math">\(N\)</span> itself: if there are twice as many bacteria, then N will grow twice as rapidly. So we have:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
\textrm{d}N/\textrm{d}t = \sigma N.\tag{1.4.1}
\end{equation}
</div>
<p class="continuation">where <span class="process-math">\(\sigma\)</span> is a constant, and <span class="process-math">\(\textrm{d}N/\textrm{d}t\)</span> is the derivative (rate of change) of <span class="process-math">\(N\)</span> with respect to time. We would have to do further experiments to find out the value of <span class="process-math">\(\sigma\text{.}\)</span> We can easily verify that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
N(t) = 1000e^{\sigma t}\tag{1.4.2}
\end{equation}
</div>
<p class="continuation">is a solution of this differential equation with the given initial condition. To do this, first calculate <span class="process-math">\(N(0)\)</span> and verify that it is the same as the number given:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
N(0) = 1000 e^{0} = 1000.\tag{1.4.3}
\end{equation}
</div>
<p class="continuation">Next, calculate <span class="process-math">\(\textrm{d}N/\textrm{d}t\)</span> and verify that it satisfies the differential equation:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation}
\frac{\textrm{d} N} {\textrm{d} t }= 1000\sigma e^{\sigma t }= \sigma 1000e^{\sigma t}  = \sigma N \textrm{d} t\tag{1.4.4}
\end{equation}
</div>
<p class="continuation">as required.</p>
<span class="incontext"><a href="sec_4-intro.html#p-11" class="internal">in-context</a></span>
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